ISSN 1472-2739 (on-line) 1472-2747 (printed) Algebraic & Geometric Topology Volume 5 (2005) 713–724 Published: 5 July 2005 713 ATG H -space structure on pointed mapping spaces Yves Félix Daniel Tanré Abstract We investigate the existence of an H -space structure on the function space, F∗ (X, Y, ∗), of based maps in the component of the trivial map between two pointed connected CW-complexes X and Y . For that, we introduce the notion of H(n)-space and prove that we have an H -space structure on F∗ (X, Y, ∗) if Y is an H(n)-space and X is of LusternikSchnirelmann category less than or equal to n. When we consider the rational homotopy type of nilpotent finite type CW-complexes, the existence of an H(n)-space structure can be easily detected on the minimal model and coincides with the differential length considered by Y. Kotani. When X is finite, using the Haefliger model for function spaces, we can prove that the rational cohomology of F∗ (X, Y, ∗) is free commutative if the rational cup length of X is strictly less than the differential length of Y , generalizing a recent result of Y. Kotani. AMS Classification 55R80, 55P62, 55T99 Keywords Mapping spaces, Haefliger model, Lusternik-Schnirelmann category 1 Introduction Let X and Y be pointed connected CW-complexes. We study the occurrence of an H -space structure on the function space, F∗ (X, Y, ∗), of based maps in the component of the trivial map. Of course when X is a co-H -space or Y is an H -space this mapping space is an H -space. Here, we are considering weaker conditions, both on X and Y , which guarantee the existence of an H -space structure on the function space. In Definition 3, we introduce the notion of H(n)-space designed for this purpose and prove: Proposition 1 Let Y be an H(n)-space and X be a space of LusternikSchnirelmann category less than or equal to n. Then the space F∗ (X, Y, ∗) is an H -space. c Geometry & Topology Publications 714 Yves Félix and Daniel Tanré The existence of an H(n)-structure and the Lusternik-Schnirelmann category (LS-category in short) are hard to determine. We first study some properties of H(n)-spaces and give some examples. Concerning the second hypothesis, we are interested in replacing cat(X) ≤ n by an upper bound on an approximation of the LS-category (see [5, Chapter 2]). We succeed in Proposition 7 with an hypothesis on the dimension of X but the most interesting replacement is obtained in the rational setting which constitutes the second part of this paper. We use Sullivan minimal models for which we refer to [6]. We recall here that each finite type nilpotent CW-complex X has a unique minimal model (∧V, d) that characterises all the rational homotopy type of X . We first prove that the existence of an H(n)-structure on a rational space X0 can be easily detected from its minimal model. It corresponds to a valuation of the differential of this model, introduced by Y. Kotani in [11]: The differential d of the minimal model (∧V, d) can be written as d = d1 + d2 + · · · where di increases the word length by i. The differential length of (∧V, d), denoted dl(X), is the least integer n such that dn−1 is non zero. As a minimal model of X is defined up to isomorphism, the differential length is a rational homotopy type invariant of X , see [11, Theorem 1.1]. Proposition 8 establishes a relation between dl(X) and the existence of an H(n)-structure on the rationalisation of X . Finally, recall that the rational cup-length cup0 (X) of X is the maximal length of a nonzero product in H >0 (X; Q). In [11], by using this cup-length and the invariant dl(Y ), Y. Kotani gives a necessary and sufficient condition for the rational cohomology of F∗ (X, Y, ∗) to be free commutative when X is a rational formal space and when the dimension of X is less than the connectivity of Y . We show here that a large part of the Kotani criterium remains valid, without hypothesis of formality and dimension. We prove: Theorem 2 Let X and Y be nilpotent finite type CW-complexes, with X finite. (1) The cohomology algebra H ∗ (F∗ (X, Y, ∗); Q) is free commutative if cup0 (X) < dl(Y ). (2) If dim(X) ≤ conn(Y ), then the cohomology algebra H ∗ (F∗ (X, Y, ∗); Q) is free commutative if, and only if, cup0 (X) < dl(Y ). As an application, we describe in Theorem 12 the Postnikov tower of the rationalisation of F∗ (X, Y, ∗) where X is a finite nilpotent space and Y a finite type CW-complex whose connectivity is greater than the dimension of X . Our description implies the solvability of the rational Pontrjagin algebra of Ω(F∗ (X, Y, ∗)). Algebraic & Geometric Topology, Volume 5 (2005) H -space structure on pointed mapping spaces 715 Section 2 contains the topological setting and the proof of Proposition 1. The link with rational models is done in Section 3. Our proof of Theorem 2 uses the Haefliger model for mapping spaces. In order to be self-contained, we recall briefly Haefliger’s construction in Section 4. The proof of Theorem 2 is contained in Section 5. Finally, Section 6 is devoted to the description of the Postnikov tower. In this text, all spaces are supposed of the homotopy type of connected pointed CW-complexes and we will use cdga for commutative differential graded algebra. A quasi-isomorphism is a morphism of cdga’s which induces an isomorphism in cohomology. 2 Structure of H(n)-space First we recall the construction of Ganea fibrations, pX n : Gn (X) → X . i pX 0 0 • Let F0 (X) → G0 (X) → X denote the path fibration on X , ΩX → PX → X. in pX n / X has been constructed. We / Gn (X) • Suppose a fibration Fn (X) X extend pn to a map qn : Gn (X)∪C(Fn (X)) → X , defined on the mapping cone of in , by setting qn (x) = pX n (x) for x ∈ Gn (X) and qn ([y, t]) = ∗ for [y, t] ∈ C(Fn (X)). • Now convert qn into a fibration pX n+1 : Gn+1 (X) → X . This construction is functorial and the space Gn (X) has the homotopy type of the nth -classifying space of Milnor [12]. We quote also from [8] that the direct limit G∞ (X) of the maps Gn (X) → Gn+1 (X) has the homotopy type of X . As spaces are pointed, one has two canonical applications ιln : Gn (X) → Gn (X ×X) and ιrn : Gn (X) → Gn (X × X) obtained from maps X → X × X defined respectively by x 7→ (x, ∗) and x 7→ (∗, x). Definition 3 A space X is an H(n)-space if there exists a map µn : Gn (X × X) → X such that µn ◦ ιln = µn ◦ ιrn = pX n : Gn (X) → X . Directly from the definition, we see that an H(∞)-space is an H -space and that any space is a H(1)-space. Recall also that any co-H -space is of LS-category 1. Then, Proposition 1 contains the trivial cases of a co-H -space X and of an H -space Y . Algebraic & Geometric Topology, Volume 5 (2005) Yves Félix and Daniel Tanré 716 Proof of Proposition 1 From the hypothesis, we have a section σ : X → Gn (X) of the Ganea fibration pX n and a map µn : Gn (Y ×Y ) → Y extending the Y Ganea fibration pn , as in Definition 3. If f and g are elements of F∗ (X, Y, ∗), we set f • g = µn ◦ Gn (f × g) ◦ Gn (∆X ) ◦ σ, where ∆X denotes the diagonal map of X . One checks easily that f • ∗ ≃ ∗•f ≃ f. In the rest of this section, we are interested in the existence of H(n)-structures on a given space. For the detection of an H(n)-space structure, one may replace the Ganea fibrations pX n by any functorial construction of fibrations p̂n : Ĝn (X) → X such that one has a functorial commutative diagram, q FF FF FF p̂n FF# Ĝn (X) X. 1 Gn (X) xx x xx x x X {xx pn Such maps p̂n are called fibrations à la Ganea in [13] and substitutes to Ganea fibrations here. Moreover, as we are interested in product spaces, the following filtration of the space G∞ (X) × G∞ (Y ) plays an important role: (G(X) × G(Y ))n = ∪i+j=n Gi (X) × Gj (Y ) . In [10], N. Iwase proved the existence of a commutative diagram p PPP PPP PPP PPP X ∪(pi ×pY ( j ) (G(X) × G(Y ))n 0 Gn (X × Y ) pp ppp p p p px pp pnX×Y X ×Y and used it to settle a counter-example to the Ganea conjecture. Therefore, in Definition 3, we are allowed to replace the Ganea space Gn (X × X) by (G(X) × G(X))n . Moreover, if p̂n : Ĝn (X) → X are substitutes to Ganea fibrations as above, we may also replace Gn (X × X) by (Ĝ(X) × Ĝ(Y ))n = ∪i+j=n Ĝi (X) × Ĝj (Y ) . We will use this possibility in the rational setting. In the case n = 2, we have a cofibration sequence, Σ(G1 (X) ∧ G1 (X)) Wh / G1 (X) ∨ G1 (X) Algebraic & Geometric Topology, Volume 5 (2005) / G1 (X) × G1 (X), H -space structure on pointed mapping spaces 717 coming from the Arkowitz generalisation of a Whitehead bracket, [2]. Therefore, the existence of an H(2)-structure on a space X is equivalent to the triviality X X X of (pX 1 ∨ p1 ) ◦ W h. As the loop Ωp1 of the Ganea fibration p1 : G1 (X) → X admits a section, we get the following necessary condition: – if there is an H(2)-structure on X , then the homotopy Lie algebra of X is abelian, i.e. all Whitehead products vanish. Example 4 In the case X is a sphere S n , the existence of an H(2) structure on S n implies n = 1, 3 or 7, [1]. Therefore, only the spheres which are already H -spaces endow a structure of H(2) space. One can also observe that, in general, if a space X is both of category n and an H(2n)-space, then it is an H -space. The law is given by X ×X σ / G2n (X × X) µ2n /X , X×X where the existence of the section σ to p2n comes from cat(X × X) ≤ 2 cat(X). Example 5 If we restrict to spaces whose loop space is a product of spheres or of loop spaces on a sphere, the previous necessary condition becomes a criterion. For instance, it is proved in [3] that all Whitehead products are zero in the complex projective 3-space. This implies that CP 3 is an H(2)-space. (Observe that CP 3 is not an H -space.) From [3], we know also that the homotopy Lie algebra of CP 2 is not abelian. Therefore CP 2 is not an H(2)-space. The following example shows that we can find H(n)-spaces, for any n > 1. Example 6 Denote by ϕr : K(Z, 2) → K(Z, 2r) the map corresponding to the class xr ∈ H 2r (K(Z, 2); Z), where x is the generator of H 2 (K(Z, 2); Z). Let E be the homotopy fibre of ϕr . We prove below that E is an H(r − 1)-space. First we derive, from the homotopy long exact sequence associated to the map ϕr , that ΩE has the homotopy type of S 1 × K(Z, 2r − 2). Therefore, the only obstruction to extend Gr−1 (E) ∨ Gr−1 (E) → E to (G(E) × G(E))r−1 = ∪i+j=r−1 Gi (E) × Gj (E) lies in Hom(H2r ((G(E) × G(E))r−1 ; Z), π2r−2 (E)). If A and B are CW-complexes, we denote by A ∼n B the fact that A and B have the same n-skeleton. If we look at the Ganea total spaces and fibres, we get: ΣΩE ∼2r S 2 ∨ S 2r−1 ∨ S 2r , F1 (E) = ΩE ∗ ΩE ∼2r S 3 ∨ S 2r ∨ S 2r , Algebraic & Geometric Topology, Volume 5 (2005) Yves Félix and Daniel Tanré 718 and more generally, Fs (E) ∼2r S 2s+1 , for any s, 2 ≤ s ≤ r − 1. Observe also that H2r (F2 (E); Z) → H2r (G1 (E); Z) is onto. (As we have only spherical classes in this degree, this comes from the homotopy long exact sequence.) As a conclusion, we have no cell in degree 2r in (G(E) × G(E))r−1 and E is an H(r − 1)-space. We end this section with a reduction to a more computable invariant than the LS-category. Consider ρX n : X → G[n] (X) the homotopy cofibre of the Ganea X fibration pn . Recall that, by definition, wcatG (X) ≤ n if the map ρX n is homotopically trivial. Observe that we always have wcatG (X) ≤ cat(X), see [5, Section 2.6] for more details on this invariant. Proposition 7 Let X be a CW-complex of dimension k and Y be a CWcomplex (c − 1)-connected with k ≤ c − 1. If Y is an H(n)-space such that wcatG (X) ≤ n, then F∗ (X, Y, ∗) is an H -space. Proof Let f and g be elements of F∗ (X, Y, ∗). Denote by ι̃X n : F̃n (X) → X the homotopy fibre of ρX : X → G (X). This construction is functorial and [n] n the map (f, g) : X → Y × Y induces a map F̃n (f, g) : F̃n (X) → F̃n (Y × Y ) such that ι̃nY ×Y ◦ F̃n (f, g) = (f, g) ◦ ι̃X n . By hypothesis, we have a homotopy section σ̃ : X → F̃n (X) of ι̃X n . Therefore, one gets a map X → F̃n (Y × Y ) as F̃n (f, g) ◦ σ̃ . Recall now that, if A → B → C is a cofibration with A (a − 1)-connected and C (c − 1)-connected, then the canonical map A → F in the homotopy fibre of B → C is an (a + c − 2)-equivalence. We apply it in the following situation: Gn (Y × Y ) Y ×Y jn Y ×Y ρn Y ×Y pn /Y ×Y ll6 l l l lll lllι̃Y ×Y l l n l / G[n] (Y × Y ) F̃n (Y × Y ) The space Gn (Y × Y ) is (c − 1)-connected and G[n] (Y × Y ) is c-connected. Therefore the map jnY ×Y is (2c − 1)-connected. From the hypothesis, we get k ≤ c − 1 < 2c − 1 and the map jnY ×Y induces a bijection [X, Gn (Y × Y )] ∼ = / [X, F̃ (Y × Y )]. n Denote by gn : X → Gn (Y × Y ) the unique lifting of F̃n (f, g) ◦ σ̃. The composition g • f is defined as µn ◦ gn where µn is the H(n)-structure on Y . Algebraic & Geometric Topology, Volume 5 (2005) H -space structure on pointed mapping spaces 719 If we set g = ∗, then F̃n (f, g) is obtained as the composite of F̃n (f ) with the map F̃n (Y ) → F̃n (Y × Y ) induced by y 7→ (y, ∗). As before, one has an isomorphism / [X, F̃ (Y )]. [X, Gn (Y )] n ∼ = A chase in the following diagram shows that f • ∗ = f as expected, F̃n (X) Gn (Y ) / Gn (Y × Y ) F̃n (f ) / F̃ (Y ) n / F̃ (Y × Y ) n f O ι̃Y n σ̃ X 3 / Y. Rational characterisation of H(n)-spaces Define mH (X) as the greatest integer n such that X admits an H(n)-structure and denote by X0 the rationalisation of a nilpotent finite type CW-complex X . Recall that dl(X) is the valuation of the differential of the minimal model of X , already defined in the introduction. Proposition 8 Let X be a nilpotent finite type CW-complex of rationalisation X0 . Then we have: mH (X0 ) + 1 = dl(X). Proof Let (∧V, d) be the minimal model of X . Recall from [7] that a model of the Ganea fibration pX n is given by the following composition, ¯ ֒→ (∧V / ∧>n V, d) ¯ ⊕ S, (∧V, d) → (∧V / ∧>n V, d) where the first map is the natural projection and the second one the canonical injection together with S · S = S · V = 0 and d(S) = 0. As the first map is functorial and the second one admits a left inverse over (∧V, d), we may use the realisation of (∧V, d) → (∧V / ∧>n V, d) as substitute for the Ganea fibration. Suppose dl(X) = r. We consider the cdga (∧V ′ , d′ ) ⊗ (∧V ′′ , d′′ )/Ir where (∧V ′ , d′ ) and (∧V ′′ , d′′ ) are copies of (∧V, d) and where Ir is the ideal Ir = ⊕i+j≥r ∧i V ′ ⊗ ∧j V ′′ . Observe that this cdga has a zero differential and that the morphism ϕ : (∧V, d) → (∧V ′ , d′ ) ⊗ (∧V ′′ , d′′ )/Ir Algebraic & Geometric Topology, Volume 5 (2005) Yves Félix and Daniel Tanré 720 defined by ϕ(v) = v ′ + v ′′ satisfies ϕ(dv) = 0. Therefore ϕ is a morphism of cdga’s and its realisation induces an H(n)-structure on the rationalisation X0 . That shows: mH (X0 ) + 1 ≥ dl(X). Suppose now that mH (X0 ) + 1 > dl(X) = r. By hypothesis, we have a morphism of cdga’s ϕ : (∧V, d) → (∧V ′ , d′ ) ⊗ (∧V ′′ , d′′ )/Ir+1 . By construction, in this quotient, a cocycle of wedge degree r cannot be a coboundary. Since the composition of ϕ with the projection on the two factors + ′ + ′′ is the natural projection, we have ϕ(v)−v ′ −v ′′ ∈ ∧ P V ⊗∧ V . Now let v ∈ V , of lowest degree with dr (v) 6= 0. From dr (v) = i1 ,i2 ,...,ir ci1 i2 ...ir vi1 vi2 . . . vir , we get X ϕ(dv) = ci1 i2 ...ir (vi′1 + vi′′1 ) · (vi′2 + vi′′2 ) · · · (vi′r + vi′′r ) . i1 ,i2 ,...,ir This expression cannot be a coboundary and the equation dϕ(x) = ϕ(dx) is impossible. We get a contradiction, therefore one has mH (X0 )+ 1 = dl(X). 4 The Haefliger model Let X and Y be finite type nilpotent CW-complexes with X of finite dimension. Let (∧V, d) be the minimal model of Y and (A, dA ) be a finite dimensional model for X , which means that (A, dA ) is a finite dimensional cdga equipped with a quasi-isomorphism ψ from the minimal model of X into (A, dA ). Denote by A∨ the dual vector space of A, graded by (A∨ )−n = Hom(An , Q) . i + We set A+ = ⊕∞ i=1 A , and we fix an homogeneous basis (a1 , · · · , ap ) of A . The dual basis (as )1≤s≤p is a basis of B = (A+ )∨ defined by has ; at i = δst . We construct now a morphism of algebras ϕ : ∧V → A ⊗ ∧(B ⊗ V ) by ϕ(v) = p X as ⊗ (as ⊗ v) . s=1 In [9] Haefliger proves that there is a unique differential D on ∧(B ⊗ V ) such that ϕ is a morphism of cdga’s, i.e. (dA ⊗ D) ◦ ϕ = ϕ ◦ d. Algebraic & Geometric Topology, Volume 5 (2005) H -space structure on pointed mapping spaces 721 In general, the cdga (∧(B ⊗V ), D) is not positively graded. Denote by D0 : B ⊗ V → B ⊗ V the linear part of the differential D. We define a cdga (∧Z, D) by constructing Z as the quotient of B ⊗ V by ⊕j≤0 (B ⊗ V )j and their image by D0 . Haefliger proves: Theorem 9 [9] The commutative differential graded algebra (∧Z, D) is a model of the mapping space F∗ (X, Y, ∗). 5 Proof of Theorem 2 Proof We start with an explicit description of the Haefliger model, keeping the notation of Section 4. The cdga (∧V, d) is a minimal model of Y and we choose for V a basis (vk ), indexed by a well-ordered set and satisfying d(vk ) ∈ ∧(vr )r<k for all k . As homogeneous basis (as )1≤s≤p of A, we choose elements hi , ej and bj such that: – the elements hi are cocycles and their classes [hi ] form a linear basis of the reduced cohomology of A; – the elements ej form a linear basis of a supplement of the vector space of cocycles in A, and bj = dA (ej ). We denote by hi , ej and P bj the corresponding elements of the basis of B = + ∨ (A ) . By developing D0 ( s as ⊗ (as ⊗ v)) = 0, we get a direct description of the linear part D0 of the differentiel D of the Haefliger model: j D0 (bj ⊗ v) = −(−1)|b | ej ⊗ v and D0 (hi ⊗ v) = 0, for each v ∈ V. A linear basis of the graded vector j b ⊗ vk , ej ⊗ vk , i h ⊗ vk , space Z is therefore given by the elements: with |bj ⊗ vk | ≥ 1, with |ej ⊗ vk | ≥ 2, with |hi ⊗ vk | ≥ 1. P Now, from ϕ(dv) = (D−D0 )ϕ(v) and d(v) = ci1 i2 ···ir vi1 vi2 · · · vir , we deduce: s (D P− D0 )(a P⊗ v) = ± ci1 i2 ···ir ai ,ai ...,air has ; ai1 ai2 · · · air i (ai1 ⊗ vi1 ) · (ai2 ⊗ vi2 ) · · · (air ⊗ vir ) 1 2 where, as usual, the sign ± is entirely determined by a strict application of the Koszul rule for a permutation of graded objects. Suppose first that cup0 (X) < dl(Y ). Algebraic & Geometric Topology, Volume 5 (2005) Yves Félix and Daniel Tanré 722 We prove, by induction on k , that in (∧Z, D) the ideal Ik generated by the elements j b ⊗ vs , s ≤ k, with degree at least 1, ej ⊗ vs , s ≤ k, with degree at least 2, is a differential ideal and that the elements hi ⊗vs , with s ≤ k and |hi ⊗vs | ≥ 1, are cocycles in the quotient ((∧Z)/Ik , D̄). Note that this ideal is acyclic as shown by the formula given for D0 . Therefore the quotient map ρ : (∧Z, D) → ((∧Z)/Ik , D) is a quasi-isomorphism. The induction will prove that the differential is zero in the quotient, which is the first assertion of Theorem 2. Begin with the induction. One has dv1 = 0 which implies (D−D0 )(as ⊗v1 ) = 0. j Therefore, we deduce D(bj ⊗ v1 ) = −(−1)|b | ej ⊗ v1 and D(hi ⊗ v1 ) = 0. That proves the assertion for k = 1. We suppose now that the induction step is true for the integer k . Taking the quotient by the ideal Ik gives a quasi-isomorphism ρ : (∧Z, D) → (∧T, D) := ((∧Z)/Ik , D) . As the elements bj ⊗ vs and ej ⊗ vs , s ≤ k , have disappeared and as cup0 (X) < j dl(Y ), we have ρ ◦ ϕ(dvk+1 ) = 0. Therefore D(bj ⊗ vk+1 ) = −(−1)|b | ej ⊗ vk+1 and D(hi ⊗ vk+1 ) = 0. The induction is thus proved. We consider now the case cup0 (X) ≥ dl(Y ) in the case dim(X) ≤ conn(Y ). We choose first in the lowest possible degree q an element y ∈ V that satisfies dy = dr y + · · · with dr (y) 6= 0 and r ≤ cup0 (X). As above we can kill all the elements ej ⊗ v and bj ⊗ v with |v| < q and keep a quasi-isomorphism ρ : (∧Z, D) → (∧T, D) := (∧Z/Iq−1 , D). Next we choose cocycles, h1 , h2 , · · · , hm , such that the class [ω], associated to the product ω = h1 · h2 · · · hm , is not zero. We choose m ≥ r and suppose that ω is in the highest degree for such a product. Let ω ′ be an element in A∨ such that hω ′ ; ωi = 1. Then, by the Haefliger formula, the differential D is zero in ∧T in degrees strictly less than |ω ′ ⊗ y|. Observe that |ω ′ ⊗ y| ≥ 2 and that the Dr part of the differential D(ω ′ ⊗ y) is a nonzero sum. This proves that the cohomology is not free. Example 10 Let X be a space with cup0 (X) = 1, which means that all products are zero in the reduced rational cohomology of X . Then, for any nilpotent finite type CW-complex Y , the rational cohomology H ∗ (F∗ (X, Y, ∗); Q) is a free commutative graded algebra. For instance, this is the case for the (nonformal) space X = S 3 ∨ S 3 ∪ω e8 , where the cell e8 is attached along a sum of triple Whitehead products. Algebraic & Geometric Topology, Volume 5 (2005) H -space structure on pointed mapping spaces 723 Example 11 When the dimension of X is greater than the connectivity of Y , the degrees of the elements have some importance. The cohomology can be commutative free even if cup0 (X) ≥ dl(Y ). For instance, consider X = S 5 × S 11 and Y = S 8 . One has cup0 (X) = dl(Y ) = 2 and the function space F∗ (X, Y, ∗) is a rational H -space with the rational homotopy type of K(Q, 3)×K(Q, 4)×K(Q, 10), as a direct computation with the Haefliger model shows. 6 Rationalisation of F∗(X, Y, ∗) when dim(X) ≤ conn(Y ) Let X be a finite nilpotent space with rational LS-category equal to m − 1 and let Y be a finite type nilpotent CW-complex whose connectivity c is greater than the dimension of X . We set r = dl(Y ) and denote by s the maximal integer such that m/r s ≥ 1, i.e. s is the integral part of logr m. Theorem 12 There is a sequence of rational fibrations Kk → Fk → Fk−1 , for k = 1, . . . , s, with F0 = ∗, Fs is the rationalisation of F∗ (X, Y, ∗) and each space Kk is a product of Eilenberg-MacLane spaces. In particular, the rational loop space homology of F∗ (X, Y, ∗) is solvable with solvable index less than or equal to s. Proof By a result of Cornea [4], the space X admits a finite dimensional model A such that m is the maximal length of a nonzero product of elements of positive degree. We denote by (∧V, d) the minimal model of Y . k We consider the ideals Ik = A>m/r , and the short exact sequences of cdga’s Ik /Ik−1 → A/Ik−1 → A/Ik . These short exact sequences realise into cofibrations Tk → Tk−1 → Zk and the sequences (∧((A+ /Ik )∨ ⊗ V ), D) → (∧((A+ /Ik−1 )∨ ⊗ V ), D) → (∧((Ik /Ik−1 )∨ ⊗ V ), D) are relative Sullivan models for the fibrations F∗ (Zk , Y, ∗) → F∗ (Tk−1 , Y, ∗) → F∗ (Tk , Y, ∗). Now since the cup length of the space Zk is strictly less than r, the function spaces F∗ (Zk , Y, ∗) are rational H -spaces, and this proves Theorem 12. Algebraic & Geometric Topology, Volume 5 (2005) 724 Yves Félix and Daniel Tanré References [1] J Frank Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. 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Belg. 4 (1997) 333–353 MathReview Département de Mathématiques, Université Catholique de Louvain 2, Chemin du Cyclotron, 1348 Louvain-La-Neuve, Belgium and Département de Mathématiques, UMR 8524, Université de Lille 1 59655 Villeneuve d’Ascq Cedex, France Email: felix@math.ucl.ac.be, Daniel.Tanre@univ-lille1.fr Received: 13 February 2005 Algebraic & Geometric Topology, Volume 5 (2005)